Caution - long-winded post ahead...
It's different, because when Jim's characters were recast originally, they were done with the intention of one person performing them full-time, until that person either retires or passes away. So far for the last fifteen years it's been Steve Whitmire (who's been with the Muppets since the early 80's and was Wembley Fraggle and Rizzo the Rat among others) playing Kermit, Ernie, The Swedish Chef, etc, since Jim's death. Other characters have been recast as well, but it's been for the same reasons - Miss Piggy and Fozzie are now played by Eric Jacobson because Frank Oz has retired from Muppets, and Statler and Waldorf are performed by Dave Goelz (who performs Gonzo, and now Statler) and Steve Whitmire (Because Richard Hunt, the original performer of Statler passed away shortly after Jim Henson did).
What Disney wants to do - and has already started doing - is have a stable of cheap puppeteers ready to play characters whenever they (Disney) want. Here's what they've done so far - Disney's movies.com website has had a Statler and Waldorf webshow on since summer, with Dave Goelz and Steve Whitmire performing. Recently, they've replaced Goelz and Whitmire with two cheaper alternatives, and the difference in the voices and acting are noticable. The Muppeteers aren't happy with this, and neither are the fans. Recently open auditions were held across the country for new Muppeteers, and the auditions were for existing characters - puppeteers were given Kermit, Fozzie and Gonzo puppets to perform with.
The end result is a cheapening of the characters. Disney wants puppeteers to be able to appear at malls and on cruises and amusement parks as the actual characters. Think about it like this - if you went to the mall to see Kermit the Frog, and it was being advertised as THE Kermit, would you want to see it performed by a random guy Disney hired or would you want it to be the real Kermit, performed by the puppeteer whos been playing the character for over fifteen years now? It would be like going to the mall to see, I dunno, Britney Spears only to find out it's a woman who, while she does a pretty good impression, isn't really Britney Spears. (I use that as an example - I realise nobody would actually WANT to go see Britney.)
Part of the magic that the Muppets has is that they've always been performed by the same person, on TV, in movies and live appearances. Disney training up replacements takes a lot of that away, cheapening the characters and a lot of people's memories of them. One of the cool things about Muppets is that although they're fictional characters, they are celebrities, in that it's a consistent performance by one puppeteer. If you see Kermit doing a talk-show appearance or on Hollywood Squares or some dang thing, it's like seeing a celebrity - it IS Kermit, you know? The whole changebale cast thing kind of ruins the illusion, as has been evidenced by the Movies.com upset. Yes, you CAN physically have multiple performers doing the characters at once, but it takes a lot away from the integrity of the character.
Of course, this is all assuming you (or anyone, really,) cares about it at all. If you don't, feel free to not sign the petition or in fact think about it any more. I care both because I'm a fan and I know some Muppet people. Also, it's another case of Disney using their money to bully a smaller organisation into getting their own way, and it's crappy and should be stopped.